If the throws and catches are assumed to occur at the same height, we can solve for the flight time, f, and substitute it into the other equations. As a result, we get some useful equations:
f = sqrt(8 * H / g)
Vv = g * f / 2
H = g * (f^2) / 2 = g * [(tau * omega)^2] / 8

These equations tell us that to double the flight time, you must double the initial throw velocity which means quadrupling the throw height.

Graph 1: Throw Height as a Function of tau and omega (assuming throw and catch positions at the same height, g = 9.81 m/s/s)

[…]

The Optimal Cascade
The shape and relative positions of the arcs are defined by the equations for P and the optimal arc equations. We now must determine the optimal endpoints of the arcs, which will be the throw and catch positions. We will not assume throws and catches are at the same height. Define the following new variables:
E = dwell distance = distance between catch position and throw position
theta = dwell angle

Assume we are given two parabolas in a cascade, and the values for Vh, F, and P are also given. We would like to find the exact throw and catch positions on these parabolas that minimizes the distance between throw and catch. (The equations start to get messy here, so I’ll just summarize the method and the results.) First, find the optimal value of theta to minimize the dwell distance, E. This is done by finding an equation for E in terms of the given quantities and theta, and setting the derivative with respect to theta, dE/d(theta) = 0. The result is:
tan(theta) = 2 * Vh * Vh / g / F .

Substituting this value back into equations for E and H, we get the following results:
E = P * cos(theta) .
This means the distance the hands must move, E, is slightly less than P but E is still always greater than the diameter of a ball, D. The result for H is:
H = {F + [P * sin(theta) * sin(theta)]}^2 / [4 * F * tan(theta)]
As mentioned earlier, if theta is small, H can be approximated with:
H = g * f * f / 8 .

There is a limit to how low you can make an optimal cascade. The balls from one arc must graze the balls from the other arc. The limit is reached when, at the instant each ball is thrown, it grazes the previously thrown ball from the other hand. Here is a summary of how to find the equation for this limit. First, write an equation for the distance between two consecutive throws from opposite hands. By setting the time derivative of this distance = 0, we find the time at which the distance between the balls is minimized and, in an optimal pattern, they are touching. By setting this time = 0, we get an equation for patterns where balls are grazing at the instant of the throw. The result, included for completeness, is: 2v(h)(F-E*cosϕ-D/2)+gτ^2/4(v(v)-gτ/4)=0″

” Average linear size of areas with uniform sign of mean curvature for elastic and elasto-plastic sheets of size L/h=250, 500 and 1,000. In a crumpled state (R<0.4R0), this size describes the characteristic size of facets and ridges in the sheet. b, The total energy of elastic and elasto-plastic sheets of size L/h=250, 500 and 1,000, scaled by 1/(L/h)1/3. Transitions in the energy of the elastic sheets at R~0.75R0 and R~0.4R0 indicate the formation of a cone and the end of a single-cone regime, respectively. The plots shown are averages of three simulations, and the yield point of the elasto-plastic sheets is Y^(1/σ)=0.01.” […]

In elasto-plastic sheets, the situation is more complicated. It is evident from Fig. 3 that their average linear facet size and energy scale similarly as a function of compression to the elastic sheets. There are however differences in the two crumpling processes, which arise at early phases of crumpling. In vertices in particular, plastic deformations appear already for R/R0 close to unity. An elasto-plastic sheet is not able to transform into a cone necessary for a folding type of initial deformations, and large numbers of vertices and ridges appear soon after crumpling begins. This becomes increasingly pronounced for increasing L/h so that the relative facet diameter then decreases as shown in Fig. 3a. It is evident from this figure that the average ridge length scales now as mean(x)/L=f(L/h)g(R/R0), where function g(z) has a power-law form in a fairly large range of the argument. It is difficult to determine by simulations a functional form for f(z). The above scaling form means however that elasto-plastic sheets of different L/h can only have the same average (relative) ridge length for different degrees of compression. Consequently, the similarity of ridge patterns found for crumpled elastic sheets does not appear in elasto-plastic sheets. The lack of such similarity is also evident in the L/h-dependent distributions of the linear facet size in elasto-plastic sheets (Fig. 4d): sheet thickness must be scaled together with the other spatial dimensions to preserve the form of the distribution. The distributions of linear facet size are now well fitted by lognormal distributions, found previously for experimental facet size”

“Pressure is an effect which occurs when a force is applied on a surface. Pressure is the amount of force acting on a unit area. The symbol of pressure is P […] Mathematically:

P = F/A or P = dFn/dA

where:

P is the pressure,
F is the normal force,
A is the area.

Pressure is a scalar quantity. It relates the vector surface element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors:

dF(n) = -dA = -PndA

The minus sign comes from the fact that the force is considered towards the surface element, while the normal vector points outwards.” (link)

A big part of why we keep things simple is because otherwise we’d have died out long ago. And our brains aren’t big enough to understand all that stuff anyway, even if there were time to figure it all out, which there’s not.

But things don’t get any simpler by us trying to make them so. Even very basic stuff usually tends to be horribly complicated once you start to think about it. Once in a while you can actually convince yourself that this existence-stuff we have going on is really quite fantastic.

About me/this blog

This blog is mainly a site where I keep track of and share some of the stuff I read and learn. Only a small subset of the posts on this blog deal with economics – I have diverse interests, and as the category cloud in the sidebar below illustrates this blog contains posts about all kinds of stuff: Mathematics, physics, statistics, geology, geography, health care and medicine, psychology, evolutionary biology, genetics, history, anthropology, archaeology, chess, …

You’re always welcome to ask questions in the comment section. New readers should be aware that the first comment someone leaves on this blog is always withheld automatically to limit spam and needs to be approved by me before it appears on the site; so your first question or comment may not appear immediately.

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Goodreads Quotes

"Happiness and its anticipation are […] proximate mechanisms that lead us to perform and repeat acts that in the environments of history, at least, would have led to greater reproductive success." (Richard D. Alexander)